Isometric Transformation Proof Using Matrix Form

[matt223]

Homework Statement

The questions asks for a proof that if a geometrical transformation $$R$$ on a three dimensional vector space with metric $$\eta$$ is length preserving, then $$R^{T}\eta R=\eta$$. Note that the summation convention is used throughout.

The transformation is given by
$$x'_{i}=R_{ij}x_{j}$$

Homework Equations

The length of a vector is determined by the metric according to
$$l^{2}=\eta _{ij}x_{i}x_{j}$$

The Attempt at a Solution

If $$R$$ is length preserving then
$$l^{2}=\eta _{ij}x_{i}x_{j} =\eta _{ij}x'_{i}x'_{j}$$
and so
$$\eta _{ij}x_{i}x_{j}=\eta _{ij}R_{ip}x_{p}R_{jq}x_{q}$$

My question is how do I get from this stage to the desired relationship $$R^{T}\eta R=\eta$$. Perhaps this is already implied by the line above? If so, how?

PS: This is my first post here - thank you for any help!

 

[fzero]

Rearrange the Rs a bit in your expression:

$$\eta _{ij}x_{i}x_{j}=\eta _{ij}R_{ip}x_{p}R_{jq}x_{q} = (R_{ip}\eta _{ij}R_{jq})x_{p}x_{q}$$

and try to rewrite $$R_{ip}\eta _{ij}R_{jq}$$ in matrix form.